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arXiv · 2004.05221

On the distribution of addition chains

Theophilus Agama
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Abstract

Addition chains are a classical construction for fast exponentiation and related computation problems. In this paper, we study a chain for a fixed integer $n$ by decomposing each generator into a \emph{determiner} and a \emph{regulator} (gap). This viewpoint leads to explicit identities for two aggregate statistics of the chain: the sum of the determiners and the sum of the chain elements. We then derive the corresponding lower bounds by using the positivity of the regulators. In parallel, we establish an identity for the reciprocal sum of the chain, showing how the harmonic profile of the chain can also be written in terms of the same gap sequence. These identities provide a unified way to compare addition chains of the same target and length. The paper concludes with a balancing problem that asks for the chain(s) that minimize the difference between the arithmetic sum and the harmonic sum, together with a structural decomposition of that optimization objective.

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Theophilus Agama. 2020-04-10. On the distribution of addition chains. https://arxiv.org/abs/2004.05221

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